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 A030132 Digital root of Fibonacci(n). 18
 0, 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9, 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9, 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9, 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Any initial pair (a(0), a(1)) of nonzero single-digit numbers enters a cycle of length 24, except for (3, 3), which enters a cycle of length 8 and (9, 9), which is immediately periodic of period length 1. - Jonathan Vos Post, Dec 29 2005 First term that differs from A004090 is a(10). In general, all terms of A004090 having one digit are the same in this sequence. - Alonso del Arte, Sep 16 2012 Decimal expansion of 12484270798876404618091 / 1111111111111111111111110 = 0.0[112358437189887641562819] (periodic). - Daniel Forgues, Feb 27 2017 LINKS Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 S. Marivani and others, Digital Roots of Fibonacci Numbers: Problem 10974, Amer. Math. Monthly, 111 (No. 7, 2004), 628. Colm Mulcahy, Gibonacci Bracelets. Marc Renault, The Fibonacci sequence modulo m Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1). FORMULA a(n + 1) = sum of digits of (a(n) + a(n - 1)). Periodic with period 24 = A001175(9) given by {1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9} a(n) + a(n + 1) = A010077(n + 4); a(A017641(n)) = 9. - Reinhard Zumkeller, Jul 04 2007 G.f. x*( -1 -x -2*x^2 -3*x^3 -5*x^4 -8*x^5 -4*x^6 -3*x^7 -7*x^8 -x^9 -8*x^10 -9*x^11 -8*x^12 -8*x^13 -7*x^14 -6*x^15 -4*x^16 -x^17 -5*x^18 -6*x^19 -2*x^20 -8*x^21 -x^22 -9*x^23 ) / ( (x-1) *(1+x+x^2) *(1+x) *(1-x+x^2) *(1+x^2) *(x^4-x^2+1) *(1+x^4) *(x^8-x^4+1) ). - R. J. Mathar, Feb 08 2013 EXAMPLE a(10) = 1 because F(10) = 55, and since 5 + 5 = 10 and 1 + 0 = 1 is the digital root of 55. MATHEMATICA digitalRoot[n_Integer?Positive] := FixedPoint[Plus@@IntegerDigits[#]&, n]; Table[If[n == 0, 0, digitalRoot[Fibonacci[n]]], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, May 02 2011 *) Table[NestWhile[Total[IntegerDigits[#]]&, Fibonacci[n], # > 9 &], {n, 0, 90}] (* Harvey P. Dale, May 07 2012 *) PROG (Haskell) a030132 n = a030132_list !! n a030132_list =    0 : 1 : map a007953 (zipWith (+) a030132_list (tail a030132_list)) -- Reinhard Zumkeller, Aug 20 2011 (PARI) a(n)=if(n, (fibonacci(n)-1)%9+1, 0) \\ Charles R Greathouse IV, Jan 23 2013 CROSSREFS Cf. A000045 (Fibonacci numbers), A010888 (digital roots), A004090, A007953, A030133. Cf. A065076. Sequence in context: A098906 A007887 A105472 * A004090 A104205 A267758 Adjacent sequences:  A030129 A030130 A030131 * A030133 A030134 A030135 KEYWORD nonn,base,easy,nice AUTHOR youngelder(AT)webtv.net (Ana) EXTENSIONS Entry revised by N. J. A. Sloane, Aug 29 2004 STATUS approved

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Last modified January 20 23:24 EST 2020. Contains 331104 sequences. (Running on oeis4.)